Differential Calculus (2017 edition)
- Related rates intro
- Related rates intro
- Related rates (multiple rates)
- Related rates: Approaching cars
- Related rates: Falling ladder
- Related rates (Pythagorean theorem)
- Related rates: water pouring into a cone
- Related rates (advanced)
- Related rates: shadow
- Related rates: balloon
Related rates: Falling ladder
You're on a ladder. The bottom of the ladder starts slipping away from the wall. Amidst your fright, you realize this would make a great related rates problem... Created by Sal Khan.
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- Did Sal use implicit differentiation in this example because there is a relationship between
x² + h² = 100)? If yes, can we change it to
h = sqrt(100 - x²)and calculate
d/dt (sqrt(100 - x²))instead?(10 votes)
- Yes you can use that instead, if we calculate
d/dt[h] = d/dt[sqrt(100 - x^2)]:
dh/dt = (1 /(2 * sqrt(100 - x^2))) * -2xdx/dt
dh/dt = (-xdx/dt) / (sqrt(100 - x^2))
If we substitute the known values,
dh/dt = -(8)(4) / sqrt(100 - 64)
dh/dt = -32/6 = -5 1/3
So, we arrived at the same answer as Sal did in this video.(22 votes)
- Can we find the time at which the ladder touches the ground?(7 votes)
- For that we would require to express height h as a function of time t. If we did this, then we just plug h=0 into the formula and solve for t. However, we lack information to produce the formula needed.
We have figured out dh/dt, which is an approximation of our formula. Using this approximation would assume that the rate of change of the height (at THAT MOMENT) stays the same or, in other words, there is no acceleration.
The ladder has forces (gravitation and friction with the wall and the floor most importantly) acting on it, hence it should have acceleration (if those forces don't balance out). We can't figure the acceleration, because there is no information about the mass of the ladder and the materials that make up the ladder and the wall.
If just for fun, you could make some reasonable assumptions about the mass and materials in question; say that it started to slide with no initial velocity and try to express h(t). But that would definitely require modifying the problem (adding data), not just adding another question.(13 votes)
- I can't seem to find an example of the rate of change of an angle of a falling ladder with respect to time?(7 votes)
- Here is how I managed to solve it in different way. You see that as time changes you cosine proportion (8/10 right at this moment) changes, it is getting bigger. At the same time your angle is decreasing so is your sine, that is the thing you want find rate of in respect to time. We have cosine and hypotenuse data. We see that whatever change occures hypotenuse is constant. That is a nice thing so we can use arccos() function to find angle at that moment so we could plug it in sine function(angle decreases over time, sine decreases). So far so good. So, arccos(8+4t/10), here 8+4t represents change of adjacent side(at that moment the value is 8) over time and 10 represents constant lenght of hypotenuse. So
sine(arccos(8+4t/10)). Also we see that our hypotenuse is 10 times bigger that radius of unit circle. So we multiply whole expression by 10 because argument of arrcos() function is making things in unit circle proportion and we have triangle that ladder is forming 10 times bigger.
Looks like this:
10*sine(arccos(8+4t/10)) = Model of our event
d/dt [ 10*sine(arccos(8+4t/10) ) ] = 10*d/dt [ sine(arccos(8+4t/10) )] =
10* cos( arccos(8+4t/10) ) * d/dt [ arccos(8+4t/10) ] =
10*cos( arccos(8+4t/10) ) * -1/sqrt( 1-(8+4t/10)^2 ) * d/dt [ 8+4t/10 ] =
finally derivative is:
10*cos( arccos(8+4t/10 ) ) * -1/sqrt( 1-(8+4t/10)^2 ) * 4/10 .
Exatly at that moment we know that adjacent side is 8 and velocity is 4ft* time. That moment means that instant, what is time at an instant? Well we think it's infinitesimally close to zero, so we substitute in derivative t=0:
10*cos( arccos(8/10) ) * -1/sqrt( 1-(8/10)^2 ) *4/10 = 8 * -4/6 = -16/3
I think key thing to understand here is that adjacent side changes over time, that is making angle do change(decrease in our case) over time. And decreasing angle means decreasing sine of that angle over time. Also, time at that moment is like frozen, but don't want to get philosophical. PLEASE comment if you see error in logic or elsewhere.(1 vote)
- how can we find the time at which the ladder loses contact with the wall?(3 votes)
- The ladder never loses contact with the wall in this example. Try to keep in mind that in these problems, the physics is simplified. We don't worry about the mass of the ladder, it never bounces, and so forth.
For the purposes of this problem, the height begins at h=6 and ends at h=0, and x is never greater than the length of the ladder, so x begins at x=8 and ends at x=10. Were the ladder to lose contact with the wall, there would be no h, and the question would become meaningless.(10 votes)
- My intuition is the speed that the ladder moves down is also 4 ft/s (same as slide outward), because the length never changes, someone explains me why it's wrong?(3 votes)
- This intuition would be correct if the length were given by x + h (because x + h = constant would imply that dx/dt = -dh/dt).
However, the length is not given by x + h according to the Pythagorean Theorem, but rather sqrt(x^2 + h^2). So constant length means that x^2 + h^2 = constant^2, which implies that 2x dx/dt = -2h dh/dt.
So the speed at which the ladder moves down is equal to the speed at which the ladder slides outward only when x = h. The ladder moves down faster than it slides outward when x > h, and the opposite occurs when x < h.
Have a blessed, wonderful day!(3 votes)
- Everything checks out mathematically but practically I have a problem. The second after t0, the bottom would have travelled 4 ft and would be at a distance 8+4=12 ft from the wall which means the entire length of the ladder is covered (plus more) and so the ladder has touched the ground.
But according to the solution, the top was at 6 ft from the ground and was falling at the rate of ~5 (less than 6) ft/sec. It doesn't look like the top would touch the ground in the next second.(3 votes)
- The velocities of the bottom and top of the ladder cannot both be constant. The instantaneous velocity of the top of the ladder is -5.3 at the instant where x=8 and dx/dt=4. Once those values change, the velocity of the top of the ladder changes as well.
Because the velocity of the top of the ladder is not constant, we cannot find the displacement simply by multiplying velocity by time.(3 votes)
- Can we use the tangent function as a relation between both x and h?(2 votes)
- We know nothing about the angles, so tangent wouldn't do us any good.(4 votes)
- Is there a set of rules or a procedure that one should keep in mind when solving a related rates problem (since they're all different), or do you just have to solve each problem intuitively?(2 votes)
- Let's say you have a rectangle and you know that side "x" is 3 feet long and side "y" is 2 feet long. Also, you know that side "x" is growing 1 foot every minute.
x = 3 ft
y = 2 ft
dx/dt = 1 ft/min
Now, how fast is side "y" growing?
The answer is that there is simply not enough information--"y" may be growing or shrinking or even remaining the same length.
However, if I tell you that the area of the rectangle is constant then there is enough information for you to determine the rate of change of side "y" with respect to time. In other words, the constant area of the rectangle acts as a constraint because:
1.) We know something about the Area (namely, that it remains constant)
2.) Both x & y are related to area via the formula Area = x*y
Now, to solve, take the derivative with respect to time of both sides, giving you:
0 = y*(dx/dt) + x*(dy/dt)
And then doing the regular arithmetic operations you get:
dy/dt = (-y/x)*dx/dt
Lastly, you just plug in the values for x, y, & dx/dt giving you:
dy/dt = -2/3 ft/min
which is how fast y is growing at the point in time when x=3 and y=2.
Does this help?(2 votes)
- How do I know which one I need to relate with??
Like how do you form the formula to set the derivative to?
I understand getting the y x and all that, but how do I get the equation to set it to d/dt?
I don't understand the relating part. How do I know which ones to relate to which?]
Thank you!(2 votes)
- When you're asked to find the rate of change of something at this very moment, it's the same as being asked to find f'(0). Since you don't care about what happens a time later/before with the height, you need to find value of the derivative at t = 0 with the given data.
Maybe it can be better understood how they relate to each other by using the explicit function notation. Let me use
b(t)for the function of the base of the triangle respect to time:
b(t) = 8 + 4t
b'(t) = 4
When you are asked to find the rate of change at this moment, t is zero, and the functions become:
b(0) = 8
b'(0) = 4
A function of the height h with respect to time can be written as:
h(t) = √[10² - (b(t))²]← Pythagoras (:
h'(t) = 1/[2*√(10² - (b(t))²]*[-2*b(t)]*[b'(t)]← Apply chain rule twice.
h'(t) = 1/[2*√(10² - 8²] * [-2*8]*← The value of
b'(t)at this moment are 8 and 4, respectively. We just need to replace them.
h'(t) = [1/2*6] * [-2*8]*← Simplify the square root in the denominator.
h'(t) = [1/6] * [-8]* = [-8*4/6]← The 2 cancel.
h'(t) = -16/3(2 votes)
- If x is increasing at 4ft/sec after one second x would be 12ft? As the ladder is only 10ft this should have seen the ladder flat on the floor.
However h is decreasing at 16/3ft per second so from the starting point in the question where h=6, after one second it would h would be 2/3rds of a foot? How can this be if x=12, the ladder would have to have stretched to over 12ft long? I guess missing an obvious point here.(2 votes)
So I've got a 10 foot ladder that's leaning against a wall. But it's on very slick ground, and it starts to slide outward. And right when it's-- and right at the moment that we're looking at this ladder, the base of the ladder is 8 feet away from the base of the wall. And it's sliding outward at 4 feet per second. And we'll assume that the top of the latter kind of glides along the side of the wall, it stays kind of in contact with the wall and moves straight down. And we see right over here, the arrow is moving straight down. And our question is, how fast is it moving straight down at that moment? So let's think about this a little bit. What do we know and what do we not know? So if we call the distance between-- let's call the distance between the base of the wall and the base the ladder, let's call that x. We know right now x is equal to 8 feet. We also know the rate at which x is changing with respect to time. The rate at which x is changing with respect to time is 4 feet per second. So we could call this dx dt. Now let's call the distance between the top of the ladder and the base of the latter h. Let's call that h. So what we're really trying to figure out is what dh dt is, given that we know all of this other information. So let's see if we can come up with the relationship between x and h and then take the derivative with respect to time, maybe using the chain rule. And see if we can solve for dh dt knowing all of this information. Well, we know the relationship between x and h at any time because of the Pythagorean theorem. We can assume this is a right angle. So we know that x squared plus h squared is going to be equal to the length of the ladder squared, is going to be equal to 100. And what we care about is the rate at which these things change with respect to time. So let's take the derivative with respect to time of both sides of this. We're doing a little bit of implicit differentiation. So what's the derivative with respect to time of x squared? Well the derivative of x squared with respect to x is 2x. And we're going to have to multiply that times the derivative of x with respect to t, dx dt. Just to be clear, this is the chain rule. This is the derivative of x squared with respect to x, which is 2x, times dx dt to get the derivative of x squared with respect to time. Just the chain rule. Now similarly, what's the derivative of h squared with respect to time? Well that's just going to be 2h, the derivative of h squared with respect to h is 2h times the derivative of h with respect to time. Once again, this right over here is the derivative of h squared with respect to h, times the derivative of h with respect to time, which gives us the derivative of h squared with respect to time. And what do we get on the right-hand side of our equation? Well the length of our ladder isn't changing. This 100 isn't going to change with respect to time. Derivative of a constant is just equal 0. So now we have it, a relationship between the rate of change of h with respect to time. The rate of change of x with respect to time. And then at a given point in time, when the length of x is x and h is h. But do we know what h is when x is equal to 8 feet? Well, we can figure it out. When x is equal to 8 feet, we can use the Pythagorean theorem again. We get 8 feet squared, plus h squared is going to be equal to 100. So 8 squared is 64. Subtract it from both sides, you get h squared is equal to 36. Take the positive square root, a negative square root doesn't make sense because then the ladder would be below the ground, it would be somehow underground. So we get h is equal to 6. So this is something that was essentially given by the problem. So now we know. We can look at this original thing right over here, we know what x is, that was given. Right now x is 8 feet. We know the rate of change of x with respect to time. It's 4 feet per second. We know what h is right now, it is 6. So then we can solve for the rate of h with respect to time. So let's do that. So we get 2 times 8 feet, times 4 feet per second, so times 4, plus 2h, is going to be plus 2 times, our height right now is 6, times the rate at which our height is changing with respect to t is equal to 0. And so we get 2 times 8 times 4 is 64. Plus 12 dh dt is equal to 0. We can subtract 64 from both sides, we get 12. 12 times the derivative of h with respect to time is equal to negative 64. And then we just have to divide both sides by 12. And so now we get a little bit of a drum roll. The derivative, the rate of change of h with respect to time is equal to negative 64 divided by 12. It's equal to negative 64 over 12, which is the same thing as negative 16 over 3, yeah that's right. Which is equal to-- let me scroll over to the right a little bit-- negative 5 and 1/3 feet per second. So we're done. But let's just do a reality check, does that make sense that we got a negative value right over here? Well our height is decreasing. So it completely makes sense that its rate of change is indeed negative. And we're done.